We present a historical account of the asymptotics of classical Goldbach representations with special reference to the equivalence with the Riemann Hypothesis. When the primes are chosen from an arithmetic progression comparable but weaker relationships exist with the zeros of $L$-functions.

We study the asymptotic behavior of a density function $$ t \to N(f; \mathcal{N}; t) = \sum_{\{\mathbf{m}=(m_1, \dots, m_n) \in \mathbb{N}^n; \mathcal{N}(\mathbf{m}) \leq t\}} f(m_1, \dots, m_n) $$ when $f:\mathbb{N}^n \rightarrow \mathbb{C}$ is a suitable multivariable complex valued multiplicative function, and the family of generalized balls $\{\mathcal{N} \le t\} \cap [0, \infty)^n$ are determined by a positive definite form on $[0, \infty)^n$. Several arithmetic applications of this work are given.

We prove the law of the iterated logarithm for discrepancies of the sequence $\{(3/2)^k x\}$.

In this paper, we study the distribution of the zeros of the derivative of the Lerch zeta-function. We indicate the zero-free regions and the positions of the trivial zeros. Also, we consider an asymptotic formula for the number of nontrivial zeros and show that almost all nontrivial zeros are arbitrarily close to the critical line.

Zeta functions of periodic cubical lattices are explicitly derived by computing all the eigenvalues of the adjacency operators and their characteristic polynomials. We introduce cyclotomic-like polynomials to give factorization of the zeta function in terms of them and count the number of orbits of the Galois action associated with each cyclotomic-like polynomial to obtain its further factorization. We also give a necessary and sufficient condition for such a polynomial to be irreducible and discuss its irreducibility from this point of view.

The study of the value distribution and the universality property of various zeta- and $L$-functions are one of interesting topics in analytic number theory. In this paper, we focus on the investigations of the polynomial Euler products or so called Matsumoto zeta-functions, and give an overview on results for this function.

Also, the facts from the personal and scientific life of Professor Kohji Matsumoto, who introduced this function, are given. It is impossible to describe briefly his all significant results, and we will focus only on the short overview of main directions of investigations.

We introduce an automaton $M$ with some conditions. The automatic sequence $\{a(n)\}_{n=0}^{\infty}$ for $M$ gives a quantity $\mu_{a}$, which is similar to a measure on the unit interval. $M$ also gives an iterated function system, hence a fractal $A$ for $M$ is determined. Fractals treated in this paper are hybrids of the Heighway dragon curve and the Lévy dragon curve. We introduce a modified iterated function system to approximate $A$ by directed piecewise linear curves $A_{k}$. We will study a relation between $\mu_{a}$ and $A_{k}$.

We first review our previous works of Arakawa and the authors on two closely related single-variable zeta functions. Their special values at positive and negative integer arguments are respectively multiple zeta values and poly-Bernoulli numbers. We then introduce, as a generalization of Sasaki's work, level 2-analogue of one of the two zeta functions and prove results analogous to those by Arakawa and the first named author.

Professor Kohji Matsumoto and the author have started our collaborative research since around the end of 1980's. We revisit/review in this paper chronologically the results and recent progress, coming from our collaborative research especially into complete asymptotic expansions associated with various zeta-functions.

We give an overview of the theory of functional relations for zeta-functions of root systems, and show some new results on functional relations involving zeta-functions of root systems of types $B_r$, $D_r$, $A_3$ and $C_2$. To show those new results, we use two different methods. The first method, for $B_r$, $D_r$, $A_3$, is via generating functions, which is based on the symmetry with respect to Weyl groups, or more generally, on our theory of lattice sums of certain hyperplane arrangements. The second method for $C_2$ is more elementary, using partial fraction decompositions.

In the paper, we consider a collection of zeta-functions associated with normalized Hecke eigen-cusp forms for the full modular group, and prove a theorem on the simultaneous approximation of a collection of analytic functions from a wide class by a collection of discrete shifts of the above zeta-functions. For this, a hypothesis on the linear independence of a certain set is applied.

We study the values taken by the Riemann zeta-function $\zeta$ on discrete sets. We show that infinite vertical arithmetic progressions are uniquely determined by the values of $\zeta$ taken on this set. Moreover, we prove a joint discrete universality theorem for $\zeta$ with respect to certain permutations of the set of positive integers. Finally, we study a generalization of the classical denseness theorems for $\zeta$.

For a given finite group $G$, the problem whether there exist infinitely many number fields $K$ with large class number and Galois group $\mathrm{Gal}(K/\boldsymbol{\mathrm{Q}}) \cong G$ is interesting and important. This problem was proved affirmatively for some groups $G$.

In this paper, we approach this problem by considering $h_KR_K$, where $h_K$ is the class number of $K$ and $R_K$ is the regulator of $K$. We prove that there exist infinitely many bicubic number fields $K$ with large $h_KR_K$. Moreover, we also prove generalization of the claim.

Recurrence formulas for poly-Bernoulli numbers and poly-Bernolli polynomials are discussed and illustrated with several examples.

Interpolated version of Kaneko-Tsumura's multiple zeta function is defined, after reviewing the interpolated Arakawa-Kaneko multiple zeta function. The relation among these two functions is clarified.

We prove an asymptotic formula for the counting function of $y$-friable numbers in a certain range, using the saddle point method. Further, we also prove certain reasonable upper bounds for the counting function of $y$-friable numbers for various ranges of $y$ whose proofs are comparatively simpler than the earlier ones. Though these bounds need not be the best estimates, these are certainly useful in applications.

In this paper we have two issues coming from the same background. The first one is to describe a certain ratio of Fredholm determinants of integral operators arising from the Riemann zeta function by using the solution of a single integral equation. The second one is to introduce a new integral operator arising from the Riemann zeta function and to study its basic analytic properties.